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A045899
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Numbers n such that n+1 and 3*n+1 are perfect squares.
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7
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0, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| It appears that a(n)=A046175(n)-A046174(n), that is, the triangular index of the n-th pentagonal triangular number minus its pentagonal index. - Jonathan Vos Post, Feb 28 2011
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REFERENCES
| A. Baker and H. Davenport, The equations 3x^2-2=y^2 and 8x^2-7=z^2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.
A. Dujella and A. Pethoe, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306.
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LINKS
| A. Dujella, The Problem of Diophantus and Davenport, References
A. Dujella, Publications of Andrej Dujella
P. Gibbs, 1,3,8,120 ... A Diophantine Problem
P. Gibbs, Diophantine quadruples and Cayley's hyperdeterminant.
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FORMULA
| a(k)=14*a(k-1)-a(k-2)+8
a[k] = ((Sqrt[3]+2)*(7+4*Sqrt[3])^k - (Sqrt[3]-2)(7-4*Sqrt[3])^k - 4)/6 - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006
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MATHEMATICA
| f[n_] := FullSimplify[((Sqrt[3] + 2)*(7 + 4*Sqrt[3])^n - (Sqrt[3] - 2) (7 - 4 Sqrt[3])^n - 4)/6]; Array[f, 18, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006
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CROSSREFS
| Equals A046184(n+1) - 1.
Essentially the same as A051047. Cf. A067900.
Sequence in context: A116008 A086302 A053129 * A165231 A004381 A166179
Adjacent sequences: A045896 A045897 A045898 * A045900 A045901 A045902
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KEYWORD
| nonn
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AUTHOR
| Andrej Dujella (duje(AT)math.hr)
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